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NEW YORK - Nearly a year after the federal rescue of the nation’s biggest banks, taxpayers have begun seeing profits from the hundreds of billions of dollars in aid that many critics thought might never be seen again.

The profits, collected from eight of the biggest banks that have fully repaid their obligations to the government, come to about $4 billion, or the equivalent of about 15 percent annually, according to calculations compiled for The New York Times.

That does not include the roughly $35 million the government has earned from 14 smaller banks that have paid back their loans.

These early returns are by no means a full accounting of the huge financial rescue undertaken by the federal government, which still faces potentially huge long-term losses from its bailouts of the insurance giant American International Group, the mortgage companies Fannie Mae and Freddie Mac, and the automakers General Motors and Chrysler. The Treasury Department could also take a hit from its guarantees on billions of dollars of so-called toxic mortgages.

But the mere hint of bailout profits for the nearly year-old Troubled Asset Relief Program has been a welcome surprise. It has also spurred hopes that the government could soon get out of the banking business.

“The taxpayers want their money back, and they want the government out of our banking system,’’ said Representative Jeb Hensarling, Republican of Texas.

In October, a financial panic was in full swing and the Treasury Department started spending roughly $240 billion to buy preferred shares from hundreds of banks that were facing huge potential losses from troubled mortgages. Bank stocks began teetering after Lehman Brothers collapsed and the government rescued AIG.

US taxpayers were told they would eventually make a modest return from these investments, including a 5 percent quarterly dividend on the banks’ preferred shares and warrants to buy stock in the banks at a set price over 10 years. But critics warned that taxpayers might not see any profits and could lose much of the investment if the assets they were buying turned out to be worthless over time.

The government bought shares in these and many other financial companies last fall, when sinking confidence among investors pushed down many bank stocks to just a few dollars a share. As the banks strengthened and became profitable, the government authorized them to pay back the preferred stock, which had been paying quarterly dividends since October.

But the real profit came as banks were permitted to buy back the so-called warrants, whose low fixed price provided a windfall for the government as the shares of the companies soared.

But all the profits taxpayers have won could still be wiped out by two deeply troubled institutions. Both Citigroup and Bank of America are still holding mortgages and other loans that were once worth billions of dollars but whose revised values are uncertain.